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Real Laminations of Cubic Polynomials on Boundaries of Hyperbolic Components

Yueyang Wang

Abstract

Milnor divides all bounded hyperbolic components of cubic polynomials into 4 types (A), (B), (C) and (D). In this article, we characterize the real laminations of cubic polynomials on the tame boundary of all bounded hyperbolic components of type (A), (B), or (C). For such maps, we prove that the real lamination is the smallest lamination which contains the real lamination of maps in the hyperbolic component and an equivalence relation generated by one characteristic equivalence class. As an application, we show that every hyperbolic cubic polynomial except type (D) is not combinatorially rigid.

Real Laminations of Cubic Polynomials on Boundaries of Hyperbolic Components

Abstract

Milnor divides all bounded hyperbolic components of cubic polynomials into 4 types (A), (B), (C) and (D). In this article, we characterize the real laminations of cubic polynomials on the tame boundary of all bounded hyperbolic components of type (A), (B), or (C). For such maps, we prove that the real lamination is the smallest lamination which contains the real lamination of maps in the hyperbolic component and an equivalence relation generated by one characteristic equivalence class. As an application, we show that every hyperbolic cubic polynomial except type (D) is not combinatorially rigid.
Paper Structure (22 sections, 43 theorems, 46 equations, 15 figures)

This paper contains 22 sections, 43 theorems, 46 equations, 15 figures.

Table of Contents

  1. Background and Main Result
  2. Polynomials and Laminations
  3. Main Results
  4. Sructure of the Article and Strategy of the Proofs
  5. External Rays and Laminations
  6. Notations
  7. External rays of Polynomials
  8. Lamination
  9. Huasdorff Limit of External Rays
  10. Cubic Polynomials and Generalized Internal Rays
  11. Mapping Scheme and Böttcher Maps
  12. Left and Right Extensions
  13. More discussions on Periodic Cases
  14. Landing Properties of Generalized Internal Rays
  15. External Classes and External Angles
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\mathcal{H}$ be a bounded hyperbolic component of type (A), (B) or (C). For any ${\bf a} \in \partial_{\mathrm{t}} \mathcal{H}$, there exists a minimal $\tau_3$-invariant equivalence relation $\Lambda({\bf a}) \subset \mathbb{S} \times \mathbb{S}$ such that In particular, we have $\lambda_\mathbb{R}(\mathcal{H})\subsetneq \lambda_\mathbb{R}({\bf a})$.

Figures (15)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 10 more figures

Theorems & Definitions (86)

  • Theorem 1.1: lamination of maps on tame boundary
  • Corollary 1.2: semi-conjugacy
  • Theorem 1.3: combinatorial rigidity for cubic polynomials
  • Theorem 2.1: Roesch-Yin roesch2008boundary
  • Theorem 2.2: Kiwi kiwi2001rationalKIWI2004207
  • Lemma 2.3
  • Lemma 2.4: continuity of external rays landing on repelling orbit
  • Theorem 2.5: Peterson--Zakeri hausdorfflimit
  • Lemma 2.6
  • proof
  • ...and 76 more